Diversity of Linear MIMO Receivers

Multimedia Communications Laboratory University of Texas at Dallas

Introduction

The gains afforded by multi-antenna (MIMO) systems in the presence of
rich fading environments has now been well known. MIMO systems can thus
be designed that allow much higher rates than single-antenna systems,
examples of which are known in the form of BLAST and its variants [1,2].

The main idea behind BLAST is successive decoding and interference
cancellation. While BLAST has been successful in giving higher rates, it
also has high computational complexity at the receiver. This complexity
arises from the difficulty of separating the multiple signal streams
that are sent simultaneously across the MIMO channel and canceling their
cross-interference.

The problem of interference cancellation existed long before the MIMO
channel became a subject of intense interest. A convenient method for
addressing interference is linear equalization, e.g. zero-forcing and
minimum mean-square error (MMSE) equalization. This work addresses
linear receivers in the context of MIMO systems (Figure 1). The basic
idea is that the interference suppression is confined to the linear
block, while it is possible to do other operations, e.g. decoding of
error control codes, on the separated streams after the linear
receiver. The idea of linear MIMO receivers has been broached in
[3,4], among others. The main contribution of our work [5] is the
analysis of the performance of this system, which has been an open
problem.

Figure 1. Linear MIMO Receiver

Results

This investigation has given rise to interesting and partially
unexpected results.

For zero-forcing receivers, the analysis as well as simulations
show that the diversity is always M-N+1, where M and N are respectively
the number of transmit and receive antennas. This is the best achievable
diversity regardless of the encoding and decoding strategy.

For the MMSE receiver, however a more interesting scenario
arises. If the input streams are separately encoded, then the diversity
of the MMSE receiver is also M-N+1, exactly the same as the zero-forcing
receiver.

However, if we allow the input streams to be encoded jointly, then the
diversity of the system has interesting and somewhat unexpected
behavior. In this case, the diversity is dependent on the spectral
efficiency that we wish to push through the system. If we desire high
spectral efficiencies, then the diversity of the MMSE receiver is still
M-N+1. However, at low enough spectral efficiencies, higher diversities
are obtained, and in fact we show that at low-enough spectral
efficiencies, a maximum diversity of MN is available.

While this work is somewhat reminiscent of the diversity-multiplexing
tradeoff of Zhang and Tse [6], we should note the important difference
that here we speak of spectral efficiency, i.e., rate, not the
multiplexing gain (prelog factor of capacity). Therefore the picture is
very different from [6].

An example of the simulations that confirm this work is provided below
in Figure 2. In this figure, M=N=2 and the curves show rates R=1,2,4,10
bits/sec/Hz. The system model involves combined spatial encoding of the
data streams on the transmit side and linear receivers. It is clearly seen that
zero-forcing has constant diversity slope regardless of the rate, while
the MMSE receiver has a rate-dependent diversity. For comparison
purposes, we also show the outage curves for the maximum likelihood
receiver, which acts as a lower bound to both linear receivers and is
known to achieve full diversity at the fixed rates considered in this
work. It is observed that at low rates MMSE performance approximates the
maximum likelihood decoder, while at high rates it diverges from the ML
performance and is closer to the zero-forcing performance.

Figure 2. Simulation results for a 2x2 system with rates R=1,2,4,10 b/s/Hz

For more details and the full analysis, the interested reader is
referred to [5].

Open Problem

This work [5] addressed the diversity of MMSE receivers in the
asymptote of low and high rate, but the complete characterization of
the diversity of MMSE receivers as a function of rate (and possibly
block length) remains an open problem at the time of the writing of
this web page.